Parameter: λ₁₃ Identified

Algebraic Equations:

σ₁₂λ₁₃-σ₂₃,
σ₁₁λ₁₃-σ₁₃

Single Door Criterion:

{1} ⊥ {3} | {}
{1} ⊥ {3,4} | {2}

Instrumental Variable:

2 | {}

Parameter: λ₂₄ Identified

Algebraic Equations:

σ₁₃σ₂₂λ₂₄-σ₁₂σ₂₃λ₂₄+σ₁₄σ₂₃-σ₁₃σ₂₄,
σ₁₂²λ₂₄-σ₁₂σ₁₄-σ₁₁σ₂₂λ₂₄+σ₁₁σ₂₄

Single Door Criterion:

{2} ⊥ {3,4} | {1}
{2} ⊥ {4} | {1,3}

Parameter: λ₃₄ Identified

Algebraic Equations:

σ₁₃σ₂₂λ₃₄-σ₁₄σ₂₂-σ₁₂σ₂₃λ₃₄+σ₁₂σ₂₄,
σ₁₂σ₁₄σ₂₂+σ₁₂²σ₂₃λ₃₄-σ₁₁σ₂₂σ₂₃λ₃₄-σ₁₂²σ₂₄

Instrumental Variable:

1 | {2}

Parameter: ω₁₁ Identified

Algebraic Equations:

-σ₁₁+ω₁₁

Parameter: ω₁₂ Identified

Algebraic Equations:

-σ₁₂+ω₁₂

Parameter: ω₂₂ Identified

Algebraic Equations:

-σ₂₂+ω₂₂

Parameter: ω₃₃ Identified

Algebraic Equations:

σ₁₃σ₂₃-σ₁₂σ₃₃+σ₁₂ω₃₃,
σ₁₃²-σ₁₁σ₃₃+σ₁₁ω₃₃

Parameter: ω₃₄ Identified

Algebraic Equations:

-σ₁₄σ₂₃²+σ₁₃σ₂₃σ₂₄+σ₁₄σ₂₂σ₃₃-σ₁₂σ₂₄σ₃₃-σ₁₃σ₂₂σ₃₄+σ₁₂σ₂₃σ₃₄+σ₁₃σ₂₂ω₃₄-σ₁₂σ₂₃ω₃₄,
σ₁₂σ₁₄σ₂₃²-σ₁₁σ₂₃²σ₂₄-σ₁₂σ₁₄σ₂₂σ₃₃+σ₁₂²σ₂₄σ₃₃-σ₁₂²σ₂₃σ₃₄+σ₁₁σ₂₂σ₂₃σ₃₄+σ₁₂²σ₂₃ω₃₄-σ₁₁σ₂₂σ₂₃ω₃₄

Parameter: ω₄₄ Identified

Algebraic Equations:

σ₁₄²σ₂₂σ₂₃²-2σ₁₃σ₁₄σ₂₂σ₂₃σ₂₄+σ₁₃²σ₂₂σ₂₄²-σ₁₄²σ₂₂²σ₃₃+2σ₁₂σ₁₄σ₂₂σ₂₄σ₃₃-σ₁₂²σ₂₄²σ₃₃+2σ₁₃σ₁₄σ₂₂²σ₃₄-2σ₁₂σ₁₄σ₂₂σ₂₃σ₃₄+2σ₁₂²σ₂₃σ₂₄σ₃₄-2σ₁₁σ₂₂σ₂₃σ₂₄σ₃₄-σ₁₃²σ₂₂²σ₄₄-σ₁₂²σ₂₃²σ₄₄+2σ₁₁σ₂₂σ₂₃²σ₄₄+σ₁₃²σ₂₂²ω₄₄+σ₁₂²σ₂₃²ω₄₄-2σ₁₁σ₂₂σ₂₃²ω₄₄,
σ₁₂σ₁₄²σ₂₂σ₂₃²-2σ₁₁σ₁₄σ₂₂σ₂₃²σ₂₄+σ₁₁σ₁₃σ₂₂σ₂₃σ₂₄²-σ₁₂σ₁₄²σ₂₂²σ₃₃+2σ₁₂²σ₁₄σ₂₂σ₂₄σ₃₃-σ₁₂³σ₂₄²σ₃₃-2σ₁₂²σ₁₄σ₂₂σ₂₃σ₃₄+2σ₁₁σ₁₄σ₂₂²σ₂₃σ₃₄+2σ₁₂³σ₂₃σ₂₄σ₃₄-2σ₁₁σ₁₂σ₂₂σ₂₃σ₂₄σ₃₄-σ₁₁σ₁₃σ₂₂²σ₂₃σ₄₄-σ₁₂³σ₂₃²σ₄₄+2σ₁₁σ₁₂σ₂₂σ₂₃²σ₄₄+σ₁₁σ₁₃σ₂₂²σ₂₃ω₄₄+σ₁₂³σ₂₃²ω₄₄-2σ₁₁σ₁₂σ₂₂σ₂₃²ω₄₄,
σ₁₂²σ₁₄²σ₂₂σ₂₃²-2σ₁₁σ₁₂σ₁₄σ₂₂σ₂₃²σ₂₄+σ₁₁²σ₂₂σ₂₃²σ₂₄²-σ₁₂²σ₁₄²σ₂₂²σ₃₃+2σ₁₂³σ₁₄σ₂₂σ₂₄σ₃₃-σ₁₂⁴σ₂₄²σ₃₃-2σ₁₂³σ₁₄σ₂₂σ₂₃σ₃₄+2σ₁₁σ₁₂σ₁₄σ₂₂²σ₂₃σ₃₄+2σ₁₂⁴σ₂₃σ₂₄σ₃₄-2σ₁₁σ₁₂²σ₂₂σ₂₃σ₂₄σ₃₄-σ₁₂⁴σ₂₃²σ₄₄+2σ₁₁σ₁₂²σ₂₂σ₂₃²σ₄₄-σ₁₁²σ₂₂²σ₂₃²σ₄₄+σ₁₂⁴σ₂₃²ω₄₄-2σ₁₁σ₁₂²σ₂₂σ₂₃²ω₄₄+σ₁₁²σ₂₂²σ₂₃²ω₄₄

Total Effect of X₁ on X₄: λ₁₃λ₃₄ Identified

Each Parameter Identified

Back Door Criterion:


{1} ⊥ {3,4} | {2}

Vanishing Ideal

Constraint Set:

σ₁₂σ₁₃-σ₁₁σ₂₃

All constraints are determinantal

Trek-separation:

{1,2} ⊥ {1,3} | ({},{1})